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Poisson Distribution: Used for modeling number of events in an interval or set if they occur randomly and independently. If N = Number of events in an interval T, Where α = Event rate  T = Average number of events per size T Interval. e.g. Failure rate for chips in a system = 2 / year For a large number of Bernoulli trials with small success rate, p, Number of successes = Poisson

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Chebychev Inequality: Beinayme Inequality: The Chebychev inequality is a special case of this with b=m, n=2

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Memoryless Random Variables: A random variable, X, is called memoryless if, for h>0 i.e. the incremental probability of x+h is independent of x. Context is meaningless. Geometric is the only memoryless discrete random variable. Exponential is the only memoryless continuous random variable.

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Gaussian Random Variable The Gaussian distribution is also called the normal distribution and is often popularly referred to as the bell curve. It is found to be a good model for random variables in many real-world systems, and has many useful properties (as we will see later in the class). m f X (x) There is no closed form for F X (x) x  

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Standard Gaussian Random Variable Note: The textbook uses  instead of G, but we will later use  for something else.

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Functions of Random Variables: If X is a random variable Y = g(X) is also a random variable. Any event {g(X) ≤ a} can be seen as a union of events in S X. This is called the equivalent event. e.g. g(x)g(x) a i j k x